Gravitational couplings and Z2 orientifolds

Nuclear Physics B 523 (1998) 465-484

ELSEVIER

Gravitational couplings and Z2 orientifolds Keshav Dasgupta a'l, Dileep E Jatkar b'2, Sunil Mukhi a,c,3 a Theoretical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India h Mehta Research Institute of Mathematics and Mathematical Physics, Chhatnag Road, Jhusi, Allahabad 221 506, India c lnstitaut voor Theoretische Fysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands Received 4 September 1997; accepted 9 February 1998

Abstract The interplay between gravitational couplings on branes and the occurrence of fractional flux in low-dimensional orientifolds is examined. It is argued that gravitational couplings need to be assigned not only to D-branes but also to orientifold planes. The fractional charges of the orientifold d-planes can be understood in terms of flux quantization of the d - 3 form potential and modified Bianchi identities. Detailed results are presented for the case of the type liB orientifold on T6/Z2, which is dual to F-theory on a complex 4-fold with terminal singularities. (~) 1998 Elsevier Science B.V. PACS: 11.25.Hf; 11.25.Mj Keywords: Orientifolds

I. Introduction Recent developments have demonstrated the importance of a variety of extended dynamical objects, branes [ 1], in string theory. One context where (Dirichlet) branes arise naturally is in the construction of orientifolds [ 2 - 4 ] , where one can sometimes think of them as twisted sector states. Another kind of object that occurs in orientifold I E-mail [email protected]. 2 E-mail [email protected]. 3 E-mail [email protected]. 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S 0 5 5 0 - 3 2 1 3 ( 9 8 ) 0 0 155-2

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constructions is called the orientifold plane, the locus of fixed points of some discrete group. Planes are usually assumed to be non-dynamical, as indeed they are at weak coupling. But it has been shown in a few contexts [5,6] that at strong coupling they can behave like dynamical objects. Another context where the distinction between branes and planes is blurred is in F-theory compactifications at constant self-dual coupling [7-9]. Here, F-theory branes move around in groups and can even produce exceptional symmetries, yet these configurations are continuously connected within the constant-coupling moduli space to the perturbative configurations of branes and planes. An essential distinction between branes and planes is that the former carry Yang-Mills gauge fields on their world-volume, and have moduli for their locations, while the latter do not. Here we want to focus on a complementary feature in which some amount of symmetry is maintained between the two types of objects. Besides Yang-Mills couplings, branes also carry gravitational couplings localized on their world-volume. As we will see, orientifold planes also carry such localized gravitational couplings, essentially because they are loci of singular curvature. This fact neatly fits in with the various observations referred to above regarding the strong-coupling behaviour of planes. An interplay between gauge and gravitational couplings is a key feature in maintaining consistency and anomaly freedom in string compactifications. Hence exploring the apparently distinct origins of the two in theories with branes and planes may tell us something quite fundamental about the underlying theory. Another, apparently unrelated, issue in orientifolds is that while branes always carry integer charge with respect to some p-form gauge field, planes can carry fractional charge in a very precise way. We will see that the presence of gravitational couplings on planes and branes together conspires to explain this fact and render these fractional charges consistent with Dirac quantization. The appearance of fractional charges on planes can be argued for many low-dimensional compactifications of M-Theory and string theory [ 10]. The most straightforward way to see this is by examining the orientifold of the type II string on Tn/Z2 where n is even for IIB and odd for IIA. There are always 16 D-branes in the vacuum and 2 n orientifold planes. Symmetry and charge conservation dictate that each plane carries 24-" units of charge. This is fractional as soon as n is greater than 4. Analogous situations occur in M-theory. In the TS/Z2 orientifold, the fact that twistedsector states are outnumbered by the fixed planes was noted and studied in Ref. [ 11 ]. A careful analysis of this situation in Refs. [12,13] revealed that as in stringy orientifolds, the planes carry fractional charge, and explained how this is in fact consistent. The origin of fractional charge in this case is as follows. The TS/Z2 compactification has 32 fixed points. Anomaly cancellation requires 16 copies of N -- 2 tensor multiplets in six dimensions. Assuming each tensor multiplet comes from a space-filling 5-brane, this would give rise to two interesting phenomena: (i) the C ~3~ /~ 18 term from the D = I1 supergravity (C ~3~ being the 3-form of M-theory) cancels anomalies locally on the brane by anomaly inflow, and (ii) a magnetic charge +1 appears on each of the 5-branes. Therefore, charge cancellation requires the planes to carry -½C~3~-field

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magnetic charge while the anomalies automatically cancel locally, on both branes and planes, by the inflow term in the lagrangian. Some more interesting cases arise in low-dimensional compactifications of string theory and M-theory. For example, consider the type II theories on T8/Z2 orbifolds and orientifolds. Type IIB theory on the T8/Z2 orientifold has 256 orientifold planes and 16 Dl-branes. Charge cancellation would require orientifold planes to carry - ~ units of / ~ , charge. Type IIA on T8/Z2 orbifold has 256 twisted sectors, but they do not contribute any massless multiplets. The massless states in the twisted sector can only arise in the RR sector since the left or right moving fermions in the NS sector give vacuum energy greater than zero. But the RR ground state in this case does not survive GSO projection [14]. However, due to the existence of a B ~ tadpole in two dimensions [15] a consistent compactification requires X/24 1-branes (fundamental type IIA strings) to condense in the vacuum, where X is the Euler characteristic of the compact manifold. For T8/Z2 the orbifold Euler characteristic is 384, and thus tadpole cancellation requires 16 type IIA strings in the vacuum. Then charge cancellation would require the fixed points to carry - ~ units of B~p charge. This lifts to M-theory and F-theory on the same orbifold. In the M-theory case the planes have - ~ units of 3-form charge while in F-theory there are really only 26 rather than 28 fixed points (we count only fixed points on the base) and these carry - ±4 units of 4-form charge. Indeed, this is in the class of T"/Z2 orientifolds referred to above, with n = 6. Note that the orbifold TS/z2 has terminal singularities, and hence requires irrelevant rather than marginal operators in order to be blown up. String and M propagation on it, however, appear to be smooth. There is also a dual pair with chiral supersymmetry in 1 + 1 dimensions. Type liB on the T8/Z2 orbifold is chiral and has potential gravitational anomalies. The 256 twisted sectors carry a total gravitational anomaly of T 64 (in units of Pontryagin numbers) and this is cancelled by the total anomaly of 256 chiral bosons from the fixed points, which is - T64 [16]. The same anomaly cancellation occurs for M-Theory on T9/Z2 (which is conjectured to be dual to liB on T8/Z2) with some crucial differences. The fixed points, 512 in number, now give chiral fermions and their anomalies cancel the anomalies coming from the untwisted sectors. Another low-dimensional case is type IIA on the T9/Z2 orientifold. This has the usual 16 D0-branes in the vacuum and 512 orientifold points. Charge cancellation would now require the orientifold points to carry - ~ units of Au charge. We will make some new observations about these low-dimensional cases later. There is an interesting relationship between Bianchi identities and flux quantization that we will exploit in this paper. The Bianchi identity of a p-form potential (in type IIA or IIB strings or in M-theory) in d uncompactified dimensions is related to charge quantization of the field in d - 4 uncompactified dimensions. As an example, the Bianchi identity of the C ~3) field of M-theory on $1/Z2 is related to the flux quantization of G (4) = d C (3) for M-theory on TS/Z2 [13]. After compactification on a circle this descends to an analogous relation for type IIA on Sl/Z2 and TS/Z2. Other examples

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include the flux quantization of G (5) = d C (4) for type liB compactified o n T6/Z2 orientifold (C (4) is the self-dual RR 4-form potential of type liB). This is related to the Bianchi identity of type liB on the T2/Z2 orientifold. Similarly, flux quantization tbr IIA on T7/Z2, liB on TS/z2 and IIA on T9/Z2 is related to Bianchi identities on T3 /Z2, T4/Z2 and TS/Z2, respectively. This paper is organized as follows. In Section 2 we argue that orientifold planes carry gravitational WZ terms, and discuss how Bianchi identities become modified in the presence of planes and branes. In Section 3 we show that the modified Bianchi identities indeed lead to consistent behaviour of branes when transported around planes even if the latter carry fractional charge, so that Dirac quantization is always satisfied. In Section 4 we discuss Wess-Zumino terms of R 4 type that are supported on planes as well as branes. In Section 5 we focus on the special case of the T6/Z2 orientifold, where 3-branes condense in the vacuum. Gravitational couplings in 3 + l-dimensional supersymmetric gauge theories have received some attention recently [ 17-19]. We interpret our results in the context of N = 4 compactifications. Finally, in Section 6 we comment on lowdimensional cases.

2. Orientifolds and fractional charges Orientifolds are constructed by gauging the world-sheet parity transformation along with some target space discrete symmetry of type II string theory. This gauging gives non-vanishing disc tadpoles. This means orientifold planes are charged with respect to the field whose disc tadpole is non-vanishing. These charges can be cancelled by inserting an appropriate number of space-filling D-branes. In the simplest case, the Z2 orientifolds, we need 16 D-branes to cancel the charge carried by orientifold planes. These orientifolds can also be understood via T-duality. Consider type IIB string theory in 10 dimensions. Orientifolding this theory in 10 dimensions gives the type I string. All other Z2 orientifolds in lower dimensions can then be understood by T-dualizing type I string theory after toroidal compactification. In 10 dimensions, the orientifold 9-plane carries charge - 1 6 with respect to the 10-form potential. This is cancelled by condensing 16 D9-branes in the vacuum, which gives the well known SO(32) gauge group of type I string theory. The orientifold 9-plane splits into two orientifold 8-planes after compactification and T-duality on a circle. These orientifold 8-planes carry charge - 8 each with respect to 9-form potential. Thus again we need 16 D8-branes to cancel the charge on the orientifold planes. A special vacuum is the one for which the D8-branes cancel the orientifold charge locally. In this case the 16 D8-branes are placed on the two orientifold planes in bunches of eight each. As we compactify further and T-dualize along the compact directions, the number of orientifold planes keeps doubling with each action of T-duality, whereas the total charge carried by all the orientifold planes remains equal to - 1 6 which is equally distributed among them. Thus for compactifications on higher-dimensional tori, orientifold planes carry fractional charge, and special vacua with local charge cancellation do

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not exist unless the compactification tori are squashed to merge orientifold planes. (It is intriguing that this occurs just at the value of uncompactified dimension (6) below which the m(atrix) theory proposal [20] starts to become problematic [21].) The first instance where fractional charges on the orientifold planes occur is the TS/Z2 orientifold compactification of type IIA string theory. Using the relation of M-theory and type IIA string theory, the same conclusion can be reached for the TS/Z2 orientifold of M-theory. In both cases, the orientifold planes carry half-integral magnetic charge with respect to the 3-form field C t3). This phenomenon was explained by Witten [ 13], who showed that the fluxes of the 4-form field strength G (4) = d C (3) are quantized in half-integral units. Whenever the G flux through a 4-cycle M of an 11-dimensional manifold Y is half integral, the first Pontryagin class Pl (Y) of Y restricted to M is an integer divisible by two but not by four. The flux which is integrally quantized belongs to G (4) - ( P l ( Y ) / 4 ) .

Since orientifolds produce vacuum charges which are cancelled by branes, the spacetime action for the orientifolds can be written as 16

I(i~ /orient = /bulk q- ~~-'~ "DBI, i=1

(2.1)

where the first term on the right-hand side is the bulk space-time action of type IIA(IIB) string theory, subject to the orientifold projection, and the second term is the sum over Dirac-Born-Infeld actions for the 16 space-filling D-branes coupled to type IIA(IIB) potentials. The first term, the bulk string theory action, is written in 10 dimensions, whereas the space-filling D-brahe actions fill the space transverse to T"/Z2. The sum is taken over 16 points on T"/Z2 where these D-branes are localised, hence in fact the second term on the RHS of (2.1) is accompanied by n-dimensional S-functions specifying locations of D-branes on Tn/Z2. We will always consider curved non-compact space, i.e. both orientifold planes and the D-brahe world-volumes are curved. As mentioned above, in curved space the D-brane world-volume has additional couplings. These are Wess-Zumino terms which are wedge products of the p-form field with powers of the curvature 2-form R. In particular, on the world-volume of a D5-brane in curved space, there is an additional Wess-Zumino term coupling the RR 2-form B to the first Pontryagin class Pl (R) [22]. One can see this by a simple anomaly inflow argument [ 23 ]. Now consider the orientifold of type IIB string theory in 10 dimensions. In this case we have an orientifold 9-plane accompanied by 16 D9-branes, leading to type I string theory. The modified Bianchi identity for the 3-form field strength due to the anomaly cancellation condition in the type I string is

dH=

½[P~(R) - p l ( F ) ] ,

(2.2)

where pl(R) is the first Pontryagin class of the spin manifold Y and is defined as ( 1/87r2)tr (R A R) and similarly Pl (F) = ( 1/87r2)tr ( F A F). For a spin manifold, Pl

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is divisible by 2, hence h = P l / 2 defines an integer cohomology class. Let us try to understand this equation from the orientifold point of view. The D9-brane world-volume action is given by

(2.3)

SD9 = SDBI q- Swz,

where Spin is the usual Dirac-Born-Infeld action and Swz is the contribution of the Wess-Zumino terms. We will not write all these terms explicitly. The relevant WessZumino terms [23] are a

f

1

*B A 1--~--~2tr(FA F)

and

±24 J['8

1

A 1--~--~5~2tr(RA R),

(2.4)

where *B is the Poincare dual of the RR 2-form B. We also have the space-time action of type liB string theory. The term relevant for our purpose is SIlB

½f H 2 + . . . ,

(2.5)

where H is the field strength of the RR sector 2-form field B. The equation of motion for *B would then be given by

dH= ~ 1

(-trFAF+~trRAR)

(2.6)

where the RHS is the contribution coming only from 16 D9-branes. Since orientifold planes are not dynamical they do not couple to F but since we are considering both D-branes as well as orientifold planes in curved space, planes can couple to R. We claim that orientifold planes contribute a further 1 tr(R A R) _ Pl 3 167r2 6

(2.7)

to the RHS of the above equation. One way to see this is the following. In the case of D-branes both the terms in (2.4) occur at the disc order with three insertions. For the orientifold plane also these terms should contribute at the same order, except that now the disc is replaced by RP 2. The first term in Eq. (2.4) requires two open-string insertions, whereas the second term has all three closed-string insertions. Since RP 2 has no boundary and open string vertices are inserted on boundaries, RP 2 does not contribute to the first term, which is equivalent 4 By the a n o m a l y inflow argument, the W Z term that actually occurs on the brane world-volume is proportional to

j

C A tr~ exp

Bp where /~(R) = 1 -- ( p l / 2 4 ) + [ ( 7 p ~ -- 4 p 2 ) / 5 7 6 0 ] + . . . suffices to take C = *B.

and Pi are the Pontryagin numbers. For our case it

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tO the statement that the orientifold planes have no open-string dynamics or they do not couple to Yang-Mills fields. Closed-string vertex insertions are in the bulk of the world-sheet and hence are allowed on R P 2. Thus the three-point vertex on R P 2, i.e. the orientifold plane, contributes a term proportional to the second term in Eq. (2.4). Both the disc and R P 2 are tree-level diagrams, and at tree level only D-branes and orientifold planes can contribute these terms. Since we already know the modified Bianchi identity as well as the contribution of D-branes, the term Eq. (2.7) has to come from the orientifold plane. We will see that this interpretation makes sense when we consider other cases, particularly the eight-dimensional example. Although the number of orientifold planes multiplies on compactification followed by T-duality, the total contribution of orientifold planes to the appropriate Bianchi identity remains the same. In other words, if C ~n) is an RR n-form, then the ~C ~n) /~ t r ( R A R)/16cr 2 term residing on the orientifold planes is equally distributed among all the orientifold planes and the total contribution of orientifold planes to the Bianchi identity for C ~") is equal to Eq. (2.7). In the case of D5-branes, Ref. [22] could not fix the sign of the Wess-Zumino term containing the Pontryagin class. Relating this term to the Bianchi identity, it is easy to see that there is a relative minus sign between the two terms occurring in (2.4). Incorporating this contribution of orientifold planes gives us the correct Bianchi identity 1

~(-trF

dH=

A F + t r R A R).

(2.8)

Another related way of seeing this is the following. In 10 dimensions, the 3-form field strength in the type I string has a kinetic term

,J

$I = ~

H A *H,

(2.9)

where (2.10)

H = d B + co3L - co3r.

Here, dw3L -- (1/167r2)trR A R and similarly for dw3r with R replaced by F. This leads to a cross term * d B A (o~3L-w3r).

(2.11)

Now let B (6) be the dual of B defined by , d B = dB (6). Then the above coupling becomes the WZ term 1

167r2

/B(6)

A(tr(FAF)_tr(RAR)

)

(2.12)

where B (6) is a (dual) RR potential in type I. From the coefficient of the above term, it is evident that the curvature terms have the right coefficient to arise from 24 9-branes. But in reality there are only 16 9-branes,

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which contribute 2/3 of the desired factor, and the remaining 1/3 is ascribed to the planes as in Eq. (2.7). Consider now compactification of type I theory on a circle followed by T-duality. This gives us type I t theory which can also be obtained from type IIA theory on an S1/Z2 orientifold. As mentioned earlier, T-duality doubles the number of orientifold planes. In this case we get two orientifold planes and (2.7) is distributed equally between them. The vacuum with local charge cancellation is the one where eight D8-branes are located on one orientifold plane and eight on the other. Let us focus only on one of the orientifold planes. The total orientifold action (modulo projection) in the vicinity of one orientifold plane is (2.13)

Sorient = aliA + SD8.

The relevant terms in this action are

l/

G2 +

Sot'ient ~ ~

×/

f.c(5) A tr(FAF)6(xg)_ 8 167.r2

1"C~5~ A tr(R16~ -2AR)6(x9),

(2.14)

where x 9 is the compact circle coordinate, the orientifold plane that we are concentrating on is localised at x 9 = 0, and *C (5) is the 5-form dual to the RR 3-form potential C ~3) in type IIA string theory. The field strength G (4) = d C (3) + ... where extra terms have to be introduced to "solve" the Bianchi identity as we discuss below. The gauge field strength F takes values in the group SO(16) whereas the curvature (R) terms all add up, so that the WZ term of a single D8-brane is multiplied by 8. Lastly, the 6-function tells us that the branes are orthogonal to x 9. The equation of motion for the field *C ~5) is given by ' (' dG (4)- 1677.2 - ~ t r R A R - t r F A F ) 6(x9),

(2.15)

where the RHS is a contribution coming entirely from the branes. We have argued that the RAR contribution from the orientifold planes is equally distributed among the planes. In the present case, a single orientifold plane contributes ~[tr(R A R)/167r216(x 9) = (Pl/12) 8(x9). Adding this contribution to the B ianchi identity of G we obtain

'('

dG(4)= 167.re ~ t r R A R - t r F A F

)

6(x9).

(2.16)

This is the analogue, for type IIA string theory on S l/Z2, of an equation derived in the second paper of Ref. [24] in the context of the strong coupling limit of this theory, namely M-theory on SJ/Z2. That equation was used in Ref. [ 13] to show that G (4) can have half-integral fluxes in M-theory. Despite the similarity in the final equation, there is an important difference between the derivation of our result and that in Ref. [24]. In the latter case, there are really no branes, just fixed planes, since there are no moduli to break E8 x E8. For string theory

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orientifolds below 10 dimensions, perturbatively there are always branes and planes, and they can be separated from each other. Hence it is essential to understand the contribution of each one separately to obtain the correct Bianchi identity in the special charge-cancelling configuration as we have just done. An essential role was played here by the gravitational coupling on the fixed planes. Now we turn to the T 2 compactification of type I string theory. This is equivalent, by T-duality, to the T2/Z2 orientifold of type liB strings. This model has been studied in great detail by Sen [ 5 ]. Here we have four orientifold planes which carry 7-brane charge - 4 . This charge can be cancelled by putting four 7-branes on the top of each orientifold plane. From what we have said earlier every orientifold plane in this case will contribute a factor ~ [tr(R A R)/167"r2]t3(x8)t3(x 9) = (pl/24)S(xS)t3(x9). This is consistent with results of Ref. [5] : when 7-branes are taken away from the orientifold plane, the plane splits into two 7-branes which are mutually non-local and also non-local with respect to the original 7-branes. The curvature term that we expect from the orientifold plane is exactly twice as much as that contributed by a single D-brane. We therefore see that each orientifold plane in this case can split into two 7-branes which share the curvature terms. Once we take this into account the Bianchi identity becomes

dG (5) = ~

,(1

~trRA R- trF A F

)

S(xS)8(x9),

(2.17)

where G (5) = d C (4) + - . . and C (4) is the self-dual RR 4-form in type IIB string theory. Although the 24 7-branes of F-theory that emerge from the non-perturbative analysis carry different (p,q) charges, hence are related to each other by SL(2, Z) S-duality, they all must carry the same world-volume term contributing to the above Bianchi identity since C (4) is SL(2, Z)-invariant. This is a nice confirmation that, in this situation, planes really can turn into branes. Compactifying further on K3, one finds a four-dimensional theory that is dual to F-theory on K3 x K3. The 24 7-branes wrapped over K3 become 24 anti-3-branes, which give rise to - 2 4 units of tadpole in the 4-form potential because of the WZ term. This is cancelled by condensing 24 fundamental 3-branes in the 4d vacuum, as predicted in Ref. [ 15]. So we apparently have 24 anti-3-branes and 24 3-branes, although the anti-branes (which are "embedded" in the 7-branes) arise from the WZ coupling and do not break extra supersymmetry. As is clear from the above discussion, this trend continues as we compactify type I string theory down to lower dimensions. The Bianchi identities for seven-, six- and five-dimensional compactifications are

trRAR-trFAF

)

dG'7)= 16~. 1 2 ( - 1~ t r R A R - t r F A F

)

S(X6)~(X7)C~(X8)C~(X9),

)

6(xS)S(x6)6(xT)6(xS)8(x9).

dG (6)=

167r2

'('

dG (8~- 16~2

~trRAR-trF

AF

6(xT)~(xS)O(x9),

(2.18)

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We will have more to say about these Bianchi identities and their relation to fractional charges in the next section. The seven-dimensional case also shows some interesting features. In this case there are eight orientifold 6-planes, along with 16 6-branes. The planes carry - 2 units of magnetic charge with respect to the RR 1-form. The WZ term involving the RR 3-form C ~3) is shared equally between the branes and planes in this case. This suggests that there may be a situation in which the orientifold planes can behave as (single) branes, and indeed there is. Compactify the seven-dimensional theory on K3 to three space-time dimensions. This is dual to M-theory on K3 × K3, for which again it is known [15] that the vacuum contains 24 condensed 2-branes. These are precisely there to cancel the 2-branes sitting "inside" the 16 6-branes and the eight 6-planes. It is intriguing that apparently static objects like orientifold planes actually contain so much dynamics.

3. M - T h e o r y on

$1/Z2 and

type IIB on

T2/Z2

In the previous section we saw how the Bianchi identity is modified in the neighbourhood of an orientifold plane. It follows from these modified Bianchi identities that the charges associated with RR p-form potentials can be fractional. Orientifold planes are charged with respect to these p-form RR potentials, and the charge fractionalization due to modified Bianchi identities is closely related to the fractional charges on the orientifold planes. In this section we will show how these fractional charges are consistent with the Dirac quantization condition in string theory and M-theory. To illustrate this we will first consider M-theory compactified on SI/z2 [24]. The Z2 action here is an orientifold action which takes C --~ - C and leaves other fields invariant. This theory has to satisfy t0-dimensional anomaly cancellation conditions at the orientifold points of S 1/Z2. The anomaly can be cancelled by putting eight space filling 9-branes at each end of the world. This gives rise to the E8 × E8 heterotic string theory with each end of the world contributing one E8 gauge symmetry. In the M-theory picture, these branes have no moduli and therefore they are stuck at the two ends indeed, they are more like static planes than dynamical branes, although one of our conclusions has been that there is not so much of a distinction between the two objects. We have already used the well-known result that the 3-form field strength H in the heterotic string theory satisfies the modified Bianchi identity 1

dH = 1--~-~2(tr R A R -- t r F A F ) ,

(3.1)

where F is the gauge field strength taking values in E8 × E8 gauge group. This equation takes quite a different form in M-theory. If we are close to one of the orientifold points on the circle, only one of the two Es gauge symmetries is visible. At the same time only half of the Pontryagin class of the curvature contributes. Thus from this point of view, the Bianchi identity is

dH= ~

1(,

K. Dasgupta et al./Nuclear Physics B 523 (1998) 465--484

>

~trRAR-trFl AF1 ,

475

(3.2)

where the subscript 1 stands for one of the E8 groups. When we lift this equation to M-theory, the 3-form field strength H goes over to the 4-form field strength G (41 to give

,(l

~trRAR-trFl

dG (41 - 167r2

AFI

)

~(xl°),

(3.3)

where x 1° = 0 is the location of the orientifold plane. Witten [ 13] observed that since R/X R is quantized in integers, the magnetic charge of the 4-form field strength G (4) is quantized in half-integers. The existence of such half-integral charge and its consistency with Dirac quantization can be established by studying the world-volume theory of a membrane [13]. To do this let us first wrap the world-volume of the membrane on a closed 3-cycle T of the 11- (or 10)-dimensional manifold. What we want to find out is what happens to the membrane path integral when we take it around a circle. The WZ coupling of the membrane world-volume theory to C (31 is given by

.p (/:..) which when transported along the circle gives

exp

(: '1 i

TxS I

G (41

.

(3.5)

/

There is another factor which contributes to the phase of the membrane path integral. This is related to the parity anomaly in the path integral over world-volume fermions, from which it follows that the interaction f C (31 on the brane world-volume is modified as

C (31 ~ C <31 + ½tr(w3,L +

(.O3,N),

(3.6)

where oJ3,L and W3,N are the Chern-Simons 3-forms associated respectively with the tangent and normal bundles to the brane world-volume. Thus the extra phase on transporting the membrane world-volume over a circle is proportional to the first Pontryagin class of the normal bundle to T x S 1 (since the tangent bundle is trivial) 167r2 j j = exp ~tTr------~--) , where dto3, N = ( 1/16qr2)tr F A F = Pl ( N ) / 2 . Thus the total phase, which must be equal to 1, is given by [ 13]

(3.7/

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It follows that G(4)/2~ has a half-integral period precisely on those cycles on which the integral of Pl ( N ) / 2 is odd. Therefore, what is really observable is not the periods of G(4)/2~, which are half-integral, but G ( 4 ) / 2 I r - Pl ( N ) / 4 , which is always integral. In general, what appears in this condition is the full Pontryagin class pj of the tangent bundle of the ambient space-time. Thus we see that the Dirac quantization condition is obeyed by G ( 4 ) / 2 ~ - P l / 4 charges. As mentioned in the previous section, orientifold planes in the TS/z2 compactification of M-theory or of type IIA string theory have half-integral magnetic charge with respect tO G (4). If we can find a 4-cycle in TS/Z2 where tr(R A R ) / 1 6 ~ 2 = pj/2 integrates to an odd integer, this would give us half-integral charge. (In this case it is the Pontryagin class of the tangent, rather than normal, bundle to the 4-manifold that contributes.) The 4-cycle which encloses an orientifold fxed point in TS/Z2 has this desired property. Although this cycle, 34/Z2, is non-orientable, its Stiefel-Whitney class w4, which is equivalent to R A R m o d 2 in the orientable case, is unity [ 13]. Consider now the T6/Z2 compactification of type IIB string theory. The modified Bianchi identity (2.17) in type IIB compactified on T2/Z2 orientifold tells us that selfdual 5-form charges are quantized as n - 1/4, where n is an integer. The solution of the Bianchi identity (2.17) relevant for this case can be written as G~5) = d C ( 4 ) + ( ~o)3,Ll _

W3,y)(~(x8)~(x9),

(3.9)

or, alternatively, as

G(5)=dC(4)-I- I----~2

trRAR-trFAF

×21 (e(x8)6(x9) _ e(x9)6(xS) ) ,

(3.10)

where e(x) is the step function. In contrast to the case of the S 1/Z2 Bianchi, in this case neither of the solutions is free of a 6-function. This is related to the fact that, in the present case, space-time does not just acquire an end-of-the-world boundary but rather ends on submanifolds of (real) codimension 2. The two solutions above differ by the addition to G (5) of the exact form d((/w3.L -- m3,r) X I ( E ( X 8 ) 6 ( X 9) -- 6 ( X 9 ) 6 ( X 8 ) ) ) .

(3.11)

The T6/Z2 orientifold compactification is the place where we expect that the charge carried by the orientifold planes is - 1/4. To measure this charge, we consider a 5-cycle, $5/Z2, which encloses the orientifold fixed point and integrate the self-dual 5-form dC ~4) over this 5-cycle. As is clear from Eq. (2.17), in the absence of gauge fields there is another contribution coming from the term l[tr(R/X R)/167r2]~(xS). Since we are only

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interested in fractional charges we can safely ignore the gauge field contribution which is always an integer. The charge of the orientifold plane, which is - 1 / 4 , is obtained by integrating the 5-form d C (4) over $5/Z2. What remains to be done, by analogy with similar manipulations in Ref. [13], is to integrate (1/16~rZ)tr(R/X R)S(xS)/4 over

ss / z2. At this point we give a more general argument which explains the existence of fractional charge when we integrate the quantity

1 tr(R A R) T----- 2"'

16qr 2

6 ( x i ) ~ ( x j ) . . . ~ ( x t,

(3.12)

over the manifold s4+q/z2. Here, m , i , j . . . . p are the relevant integers and q is the number o f delta-functions. For the T6/Z2 case, m = 2, i = 5 and q = 1. S 4+q is defined by

S4+q : x ~ + x 2 + . - . + x 4+q 2 + x~+q = 1.

(3.13)

A n d s4+q/z2 is defined modding out with the antipodal map xi ~ - x i for all i. Now the integration is simple to perform. The 6(xi),3(xj) . . . . factors fix us at the locus xi = 0, xj = 0 . . . . . This locus is a section of s4+q/z2 which is nothing but $4/Z2 because there are q delta-functions. It is well known that $4/Z2 can be naturally embedded in S4+q/Z2. This embedding corresponds to setting q coordinates of s4+q/z2 tO zero. Integrating out the delta-function precisely implements this action. 5 Therefore, now we only have to integrate ( 1 / 2 " ) [tr(R A R ) / 1 6 ~ -2] over $4/Z2. The quantity A - ( 1 / 1 6 7 r Z ) t r ( R A R ) is congruent modulo two to the Stiefel-Whitney class w4. By a standard computation [25] one can show that

f w4 s4/ z2

1 mod2.

(3.14)

Together with the 1/2 m factor, this would point to the existence of fractional fluxes for the corresponding fields. For the T6/Z2 example considered we see that the period of the 5-form G (5) is fractional precisely on those 5-cycles on which the integral of ±4 [ tr( R A R)/16"rr 2 ] 6 ( x 5 ) contributes the compensating fraction, so that the total charge is effectively integer and the Dirac quantization condition is then satisfied by the charges

5 The simplest geometrical way to think of it is that the boundary of a ball in three dimensions is S2, and its equator is SI. A Z2 modding will make the equator S 1/Z2. So if there is a quantity to be integrated over $2/Z2 with a delta-function along say the z direction, it will reduce to an integral over SI/Z2. This also seems like $2/Z2 being represented as a "fibration" over S I/Z2 with a fibre S I. Since the Z2 action has done nothing to the fibre the integral of the delta-function will be just 1.

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of the field G ~5) - ¼[tr(R A R)/16¢r216(x). 6 The crucial thing that has entered into the discussion is that the object which we want to integrate has support only a $4/Z2 submanifold of s4+q/z2. It is interesting to consider whether other physical effects are related to the occurrence of the fractional fluxes that we have been discussing. For example, fractional fluxes similar to those discussed above would appear in other orientifolds based on higher discrete groups than Z2 and correspondingly with lower supersymmetry. However, there do seem to be some important distinctions between the situation discussed in the context of SJ/Z2 orientifolds [24] and the more general cases considered here. In the former case, one can write the modified Bianchi identity at an orientifold plane as a completely non-singular boundary condition, while in the latter cases it needs to be expressed in a singular form. Presumably related to this is the fact that the former case has wider applicability: besides orientifold planes, fractional flux for a 4-form field strength can occur in M-theory compactifications on any complex 4-fold with a 4-cycle whose first Pontryagin class is a multiple of 2 but not 4. It remains to be seen whether charges of the type 1/2 n for field strengths G (n+3) can be realized in smooth compactifications, but from the present considerations this does not seem likely. In the same vein, p-branes for p ~> 3 do not have anomalies with the right discrete ambiguity Z2~-, to play a role similar to the parity anomaly on the 2-brane. It remains to be understood what exactly happens to the parity anomaly on the 2-brane when it is T-dualized to a higher brahe. Comments in this direction appear in Refs. [27-29] but a careful analysis remains to be carried out. In this section we have seen how the fractional charges on the orientifold planes, which could potentially be inconsistent with the Dirac quantization condition, actually conspire to give consistent results in the case of T"/Z2 orientifolds with n /> 5. In subsequent sections we examine other aspects of gravitational couplings and some details of lowdimensional orientifolds.

6 In the case of $5/Z2, there is another way to show how G ~5) - ¼ltr(RA R)/16~r216(x) satisfies the Dirac quantisation condition. To see this we use the fact that S5 is a generalized Hopf fibration over CP 2 with S I fibre. The antipodal map which takes S5 to SS/Z2 acts trivially on CP 2 but it halves the volume of the fibre. Therefore, $5/Z2 is also an S 1 fibration over CP 2. The difference between these two bundles is that the Chern class of the latter is double of that of the former. To evaluate ¼[tr(R A R)/16~ -216(x5), one first integrates along the fibre to reduce the top form on $5/Z2 to the top form on the base, i.e. CP 2. Integration along the fibre can be done if the cohomology classes have compact support in the vertical direction [26]. Since the Chern class is doubled, integration along the fibre gives 1 t r ( R A R) f6(xS)dx 4 167r2 cp2 J sI

5

1 tr(RA R) 2 167r2 Cl,2 "

The RHS of the equation can be written in terms of the Pontryagin class of CP 2 as p l ( R ) / 4 . Thus the integration along the fibre gives the top form on CP 2 which is one-quarter of its first Pontryagin class. Since the integral of p l ( R ) over CP 2 is equal to 3, integrating it on CP 2 one obtains the contribution of the curvature terms, which is equal to 3/4.

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4. R 4 W e s s - Z u m i n o terms on high-dimensional branes and planes We have seen that R 2 couplings of Wess-Zumino type appear on orientifold planes as well as branes. Here we will show that the same is true for R4 couplings, though for obvious reasons these can only appear on p-branes and p-planes for p >~ 7. In 10 dimensions, the type I string has a term B A X8 which plays an essential role in the Green-Schwarz anomaly cancellation mechanism. Here X8 is the 8-form XS

_

1

( l_~_t r R 4 +

(2~r) 4 \ 1 9 2

1

7-~ (tr

2_ l R2 )2 ) = 1 _ ~-~ (Pl) ~-~P2.

(4.1)

The first Pontryagin class Pl was defined earlier, and the second Pontryagin class is given by P2-

(27/.)41 ( - 1 tr R 4+1 ~5(trR2)2 ) ' ~

.

(4.2)

As before, we decompose this term into the contribution from the bulk, the branes and the orientifold plane, all of which are of course coincident in l0 dimensions. No term of the above form is present in type IIB in 10 dimensions. The contribution on a single 9-brane, which we denote B A B8, is extracted from the anomaly inflow formula, which leads to Bs

= 3@0(½(pl)2 - /p2).

(4.3)

This can be conveniently recast in terms of X8 and tr R4, and we find 1 (2 B8 = i-6 _~X8

1 trR 4 "~ 480 ( - ~ 4 j •

(4.4)

Since we require that the contribution from 16 branes plus that from the plane must provide the total R 4 term, we have 16B8+P8 = X8, where BAP8 is the plane contribution. P8 is found to be

1X 1 tr R4 P8 = ~ 8 + 480 (27r) 4"

(4.5)

The first term has the property that the ratio of brane and plane contributions is g2 and .~, as for the R 2 interactions discussed earlier. But this time there is an additional polynomial which instead contributes with opposite signs. An interesting consequence can be deduced from the fact that the first terms occur in the above ratio. Consider type I string theory propagating in a six-dimensional spacetime times a 4-manifold such as K3 (or an ALE space) with non-zero signature. For such a configuration there is a six-dimensional term induced by integrating B A Xs over the 4-manifold. X8 contains a term proportional to (R A R) 2 and since the integral of R A R over the 4-manifold is non-zero, the induced six-dimensional term is proportional to B A R A R. Thus we have a six-dimensional configuration of branes and planes (the original 9-branes and 9-planes wrapped on the 4-manifold) which carry Wess-Zumino

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terms of the type B A R A R. From Eqs. (4.3) and (4.5), using the fact that the trR 4 terms vanish in this case, we see that the ratio of brane and plane contributions is g2 and ±3' Thus in six dimensions, even quite general wrapped branes and planes (and not just fundamental D5-branes and orientifold 5-planes) carry WZ terms in the 3, i ratio. The situation will be different if we compactify on an 8-manifold, since here the extra tr R 4 terms will be visible. One might think that the presence of these extra terms contradicts the fact that a 7plane can split into a pair of 7-branes. However, the R 4 terms on 7-branes and 7-planes are of the form ~AB8 and q~APs, respectively, where O~is the RR scalar. When 7-planes split into ( p , q ) 7-branes, since O~ is not SL(2, Z) invariant one cannot say what the (p, q) branes should carry. In this respect the situation is similar to that for the 8-form charge carried by 7-planes and 7-branes, which according to Ref. [5] does not split additively because of the non-Abelian nature of the monodromy. This is in contrast to the R 2 term, where the RR potential C (4) that appears is SL(2, Z) invariant, and the term splits additively.

5. Gravitational couplings in 3 + 1-dimensional gauge theory It has been observed that certain supersymmetric gauge theories in 3 + 1 dimensions have partition functions which are modular under SL(2, Z) with non-trivial weight. The resolution to this apparent failure of exact SL(2, Z) invariance, or modular anomaly, is that these theories have specific couplings to gravity which produce (cancelling) modular anomalies. Here we will realize the relevant gauge theories on world-volumes of 3-branes, and will investigate the relationship between the gravitational couplings required for consistency of gauge theories and those generated by branes and planes on their world-volumes. Consider N = 4 super-Yang-Mills in 3 + I d. This is the world-volume field theory of a 3-brane. It can be considered to be topologically twisted (namely, the physical and twisted theories are equivalent) when written on fiat space-time or (after Euclideanization) on hyper-K~ihler 4-manifolds. As we have seen, the gravitational Wess-Zumino couplings on the 3-brane world-volume are known to be 1 1 16rr----7 2--~tr (C(°) R A R) ,

(5.1)

where C {°) is the RR scalar of type liB. However, this coupling is not SL(2, Z) invariant or even covariant. To discover the correct extension of the above coupling, we need to realize a known vacuum of string theory in terms of condensed branes. The appropriate vacuum in this case is the orientifold of type liB on T6/Z2, which has already made an appearance above. This vacuum has N = 4 space-time supersymmetry, and its gauge sector is an N = 4 super-Yang-Mills theory. In this way of describing the vacuum, there are 16 3-branes along with 64 orientifold planes. As far as world-volume gravitational couplings

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are concerned, we have argued that the planes carry half the fraction carried by the branes, so that to find the contribution on a single brane world-volume, we need to divide relevant terms in the space-time action by 24. In Ref. [ 17 ] it is observed that the space-time R 2 coupling in 4d N = 4 compactifications is, at tree level, 7 proportional to tr (C(°)R A R + e-~bR A *R).

(5.2)

This is argued in Ref. [17] to be corrected by the replacement

( l o g 7](I") 24 ~ 27ri J '

C (°) ---+Re \

(1og (,)24)

e - ~ ~ Im \

21ri

.] ,

(5.3)

where r -- C (°) + ie -~. In the limit of constant dilaton and axion, the action obtains a contribution depending only on the topological invariants X (the Euler characteristic) and 0- (the signature) of the space-time 4-manifold

- ( X - 30") l°g~ 12 - (X + 3o') I°g# 12"

(5.4)

It follows that each brane carries ~4 of this term. To leading order and considering only the term involving o-, this precisely coincides with Eq. (5.1), given that 0- = P l / 3 where pl is the correctly normalized first Pontryagin class. The full gravitational coupling on the brane, to second order in derivatives, is thus - ¼ ( 2 X - 3o-) logn - ¼(2x + 30-) log#.

(5.5)

One way to check that this is correct is to note than on hyper-K~ihler manifolds, the modular anomaly from this must cancel that coming from the gauge partition function, for which we have the result [30]

Zgauge (-!)=7"x/2Zgauge(7"

).

(5.6)

To check cancellation of the modular anomaly, we examine the SL(2, Z ) transformation law for Eq. (5.5) after setting o-= - 2 g which is the case for hyper-K~ihler manifolds. Thus we need to know how the term in the functional integral Zgra v =

exp( - X log r/) = r/-x

( 5.7 )

transforms. Using 77(-1/7-) = 7-1/27(7-) we find that the gravitational contribution to the modular anomaly is

Zgauge (--~)=T-X/2Zgauge(7-),

(5.8)

which exactly cancels the gauge contribution.

7 Our conventions differ slightly from those in Ref. [171, since we want to make the anomaly term purely holomorphic rather than anti-holomorphic.

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6. Some issues concerning fractional charge in dimensions d < 3 Although not specifically related to gravitational couplings on branes and planes, there is a curious situation in which gravitational anomalies turn into fractional charges on planes upon compactification. We discuss this below and explain how chirai supersymmetry in this problem cures an apparent paradox. Consider the orientifold of type IIA o n T 9 / Z 2 . By T-duality, this vacuum, where all of space is compactified, is realized with 16 0-branes located at points in the internal torus (of course, in such low dimensions the concept of moduli space is not strictly appropriate). The 29 = 512 orientifold points each carry a charge - ~ i with respect to the RR 1-form. This vacuum may be considered a limit of M-theory on T9/Z2 to two space-time dimensions, further compactified on a circle, as the circle shrinks. However, in the M-theory case [ 11,14,16] one expects 512 chiral fermions to appear in the twisted sector, located symmetrically at the 512 fixed points. Thus it would appear that on compactification of the M-theory orientifold on a further circle, 512 chirat fermions in 1 + 1 non-compact dimensions must suddenly turn into 16 D0-branes, while the gravitational anomaly carried by each of the 29 fixed planes (which are really fixed lines) turns into -3~ units of 1-form charge. The fermions in 1 ÷ 1 dimensions were forced to sit at the fixed points to bring about local gravitationalanomaly cancellation, so it is hard to understand how they go over into 16 objects in 0 + 1 dimensions which apparently cannot bring about local 1-form charge cancellation. The resolution to this lies in the supersymmetry of this problem. In 1 + 1 dimensions, the above orientifold of M-theory has (0,16) chiral supersymmetry. The algebra is

{Q~_, Q J_} = 6i.iP-.

(6.1)

The chiral fermions which appear as twisted sectors are singlets of supersymmetry, which means they have + chirality in these conventions, and hence are annihilated by both sides of the algebra by virtue of the Dirac equation P_~p+ = 0. On compactification to 0 + 1 dimensions, we end up with a supersymmetric quantum mechanics that is also chiral, in the sense that now the supercharges Oi satisfy

(Qi, Qj} = 6ij(P - Z ) ,

(6.2)

where Z is a central charge. D0-branes are BPS, which means they are annihilated by the RHS of this algebra. Thus in fact the 0-branes propagating in the fully compactified space are singlets of the residual supersymmetry, rather than multiplets with 32 states (16 bosonic, 16 fermionic) as they are in higher dimensions. As a result, in type IIA o n T9/Z2 the twisted sector of the orientifold can be thought of as being made up of 512 supersymmetry singlets, and local gravitational anomaly cancellation goes directly over into local charge cancellation.

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7. Discussion We have seen that curved orientifold planes carry gravitational couplings of WZ type. Quite plausibly they also carry other types of gravitational couplings such as R 4 terms (not of Wess-Zumino type) or their dimensional reductions. This would be interesting to investigate, along with the possible appearance of similar terms on curved D-branes. It has also been argued here that fractional fluxes can consistently be carried by orientifold planes. The general conclusion from this discussion would be that such phenomena as Dirac quantization can be modified by suitable (topological) gravitational couplings, as first noted in Ref. [ 13]. While all this adds some insight into the fascinating interplay between gauge and gravitational interactions in string and M-theory, a deeper understanding of this interplay would be desirable. Also, it would be interesting to understand gravitational couplings in N = 2 supersymmetric gauge theories [ 19] from the point of view of 3-branes in suitable backgrounds.

A cknowledgements We acknowledge helpful discussions with I. Biswas, A. Dabholkar, R. Dijkgraaf, M. Douglas, S.E Hassan, C. Imbimbo, K.S. Narain, N. Nitsure, N. Raghavendra, A. Sen, W. Taylor, E. Verlinde, H. Verlinde and E. Witten. D.J. would like to thank the Tata Institute of Fundamental Research for hospitality. S.M. is grateful for support from the Nederlandse Organisatie Voor Wetenschappelijk Onderzoek and the hospitality of Herman Verlinde and the Universiteit van Amsterdam.

References 111 J. Polchinski, Phys. Rev. Lett. 75 (1995) 47, hep-th/9510017. 121 A. Sagnotti, in: Non-perturbative Quantum Field Theory, Cargese, 1987, eds. G. Mack et al. (Pergamon Press, Oxford, 1988). 131 P. Horava, Phys. Lett. B 231 (1989) 251. 141 J. Dai, R.G. Leigh and J. Polchinski, Mod. Phys. Lett. A 4 (1989) 2073. 151 A. Sen, Nucl. Phys. B 475 (1996) 562, hep-th/9605150. 61 D. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, hep-th/9609070. 71 K. Dasgupta and S. Mukhi, Phys. Lett. B 385 (1996) 125, hep-th/9606044. 8] C. Ahn and S. Nam, Compactifications of F-theory on Calabi-Yau three-folds at constant coupling, hep-th/9701129. 191 D.P. Jatkar, Non-perturbative enhanced gauge symmetries in the Gimon-Polchinski orientifold, hepth/9702031. 1101 J. Polchinski, S. Chaudhuri and C.V. Johnson, Notes on D-branes, hep-th/9602052. I11] K. Dasgupta and S. Mukhi, Nucl. Phys. B 465 (1996) 399, hep-th/9512196. 1121 E. Witten, Nucl. Phys. B 463 (1996) 383, hep-th/9512219. 113] E. Witten, Flux quantization in M-theory and the effective action, hep-th/9609122. 1141 A. Sen, Nucl. Phys. B 474 (1996) 361, hep-th/9604070.

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K. Dasgupta et aL /Nuclear Physics B 523 (1998) 465-484

1151 S. Sethi, C. Vafa and E. Witten, Constraints on low-dimensional string compactifications, hepth/9606122. [16] K. Dasgupta and S. Mukhi, Phys. Lett. B 398 (1997) 285, hep-th/9612188. [171 J.A. Harvey and G. Moore, Five-brane instanton and R2 couplings in N = 4 string theory, hepth/9610237. 1181 J.A. Harvey and G. Moore, Exact gravitational threshold correction in the FHSV model, hep-th/9611176. [19l E. Witten, On S-duality in abelian gauge theory, hep-th/9505186. [201 T. Banks, W. Fischler, S.H. Shenker and L. Susskind, Phys. Rev. D 55 (1997) 5112, hep-th/9610043. 1211 N. Seiberg, New theories in six dimensions and matrix description of M-theory on T5 and TS/Z2, hep-th/9705221. [221 M. Bershadsky, V. Sadov and C. Vafa, Nucl. Phys. B 463 (1996) 420, hep-th/9611222. [231 M.B. Green, J.A. Harvey and G. Moore, Class. Quant. Gray. 14 (1997) 47, hep-th/9605033. 1241 P. Horava and E. Witten, Nucl. Phys. B 460 (1996) 506, hep-th/9510209; Eleven-dimensional Supergravity on a manifold with boundary, hep-th/9603142. 1251 J.W. Milnor and J.D. Stasheff, Characteristic Classes, Section 4 (Princeton University Press, 1974) p. 37. [261 R. Bott and L. Tu, Differential Forms in Algebraic Topology (Springer, Berlin). 1271 N. Seiberg, Phys. Lett. B 384 (1996) 81, hep-th/9606017. 1281 N. Seiberg, Phys. Lett. B 388 (1996) 753, hep-th/9608111. [291 N. Seiberg and E. Witten, Gauge dynamics and compactification to three dimensions, hep-th/9607163. [301 C. Vafa and E. Witten, Nucl. Phys. B 431 (1994) 3, hep-th/9408074.